The accelerating effect of coarse-mesh rebalancing on the low-order Chebyshev polynomial iterations to obtain the fundamental eigenvector of large homogeneous linear systems associated with elliptic partial-differential equations is mathematically analyzed. Coarse-mesh rebalancing is shown to have a positive accelerating effect if one of the following conditions is met: (a) the weighting vectors are not contaminated with high eigenvector components, (b) Galerkin's weighting vectors are used, or (c) the non-Galerkin weighting vectors are similar to the trial vectors. As another interesting result, it is shown that the overshooting effect is related to the fourth and higher eigenvector components that have spatially odd parities. If the above condition, (c), is met, there is no overshooting; otherwise, the acceleration effect with non-Galerkin weighting vectors is unpredictable.